I'm an Aggie who couldn't do this when I was in school. Sure can't splain it now.

I haven't done this stuff in many years, but unless I'm missing something it appears to be straight forward.
It's based on the fact that the integral of "e to the -kt power" with respect to "t"  is "-k times (e to the -kt power)".
Using that concept, each of the three terms in the first line translates into the respective terms in the second line.
Hard to explain better here because I don't have fancy fonts to write equations

Does that help?

ETA:  That's why you see each term in the result divided by the same number used in the exponent in the integral.  You have to account for the "-k times" factor stated above.

And then when you evaluate at 0 and infinity you get the final result. All the exponents evaluate to "1" for t=0 and "0" for t=infinity.

Each term in the brackets is the result of a u-substitution.  In order from left to right, the substitutions you would use are u=-4*lambda*t, u=-3*lambda*t, and u=-2*lambda*t.

First step is to integrate the terms separately most of the time.

The sum of the integrals equals the integral of the sums.

If I remember right, what you seek is called the additive property of Intergals, what brickeye said... if you want to youtube it for more examples.